Optimal ordering policies using a discounted cash-flow analysis when stock dependent demand and a trade credit is linked to order quantity

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1 Amrican Jr. of Mathmatics and Scincs Vol., No., January 0 Copyright Mind Radr Publications Optimal ordring policis using a discountd cash-flow analysis whn stock dpndnt dmand and a trad crdit is linkd to ordr quantity Nita H. Shah and Amisha R. Patl Dpartmnt of Mathmatics, Guarat Univrsity, Ahmdabad U.V.Patl Collg of Enginring, Ganpat Univrsity, Khrva, Mhsana 8007 Guarat, India. Corrsponding Author: Nita H. Shah and Amisha R. Patl nitahshah@gmail.com Abstract In this articl, th purchasr s optimal pricing and ordring policy is dvlopd whn units in invntory ar subct to dtrioration at a constant rat. h dmand is assumd to b stock dpndnt and th supplir offrs to th purchasr diffrnt trad crdits. W solv th invntory problm by using a discountd cash-flow (DCF) approach, charactriz th optimal solution, and obtain som thortical rsults to find th optimal ordr quantity and th optimal rplnishmnt tim. Numrical xampls ar givn to illustrat th proposd modl. Snsitivity analysis for stock dpndnt paramtr and dtrioration rat is carrid out to driv managrial in signts. Kywords: Invntory, dtrioration, stock dpndnt dmand, discountd cash flow, trad crdit. Introduction In th traditional EOQ modl, it is assumd that th purchasr must pay to th supplir for th itms as soon as th itms ar rcivd. Howvr, in today s businss transactions, it is mor and mor common to s that th supplir provids a prmissibl dlay in paymnts or a pric discount to th purchasr if th ordr quantity is gratr than or qual to a prdtrmind quantity. o th supplir, a prmissibl dlay in paymnts or a pric discount is an fficint mthod to stimulat mor sals. In practic, th supplir is willing to offr th rtailr a crtain crdit priod without intrst during th prmissibl dlay priod to promot markt comptition. Bfor th nd of th trad crdit priod, th rtailr can sll th goods and accumulat rvnu and arn intrst. A highr intrst is chargd if th paymnt is not sttld by th nd of th trad crdit priod. Convrsly, to th purchasr, th supplir s trad crdit rducs his/hr purchas cost. Goyal (985) usd th avrag cost approach to stablish an EOQ modl and analyz th ffct of trad crdit on th optimal invntory policy. Aggarwal and Jaggi (995) xtndd Goyal s (985) modl to allow for an xponntial dtrioration rat undr th condition of prmissibl dlay in paymnts. Jamal, Sarkr, and Wang (997) thn furthr xtndd th modl to allow for shortags. Mor rlatd articls can b found in th work by Arclus, Shah, and Srinivasan (00), Jabr (007), Sana and Chaudhuri (008), tc. All ths modls assum that supplirs offr a dlayd paymnt priod to rtailrs, but th rtailrs fail to offr th dlayd paymnt to its customrs. Chung and Huang (007) continud to amnd Huang (006) to propos an EOQ modl for dtriorating itms undr th two-lvl trad crdit policy. Jaggi, Goyal, and Gol (008) stablishd an EOQ modl undr a two-lvl trad crdit policy with crdit linkd dmand. In contrast, ng and Chang (009) modifid Huang (007) by rlaxing th assumption that th trad crdit priod offrd by th supplir is longr than th trad crdit priod offrd by th rtailr. It is worthwhil noting that all ths modls discussd optimal invntory stratgy undr 7

2 Nita H. Shah and Amisha R. Patl two-lvl trad crdit from th prspctiv of th buyr or th supplir only. Subsquntly, svral modls concrning th intgratd invntory modl with trad crdit policis hav continud to b proposd by Sarkr, Jamal, and Wang (000), Jabr and Osman (006), Yang and W (006), Chn and Kang (007), Su, Ouyang, Ho, and Chang (007), and Ho, Ouyang, and Su (008).Su t al. (007) assumd that th supplir and th rtailr adoptd a two-lvl trad crdit policy. Howvr, Subsquntly, svral modls concrning th intgratd invntory modl with trad crdit policis hav continud to b proposd by Sarkr, Jamal, and Wang (000), Jabr and Osman (006), Yang and W (006), Chn and Kang (007), Su, Ouyang, Ho, and Chang (007), and Ho, Ouyang, and Su (008).Sut al. (007) assumd that th rtailr obtaind a longr trad crdit priod from th supplir and providd a rlativly shortr trad crdit priod to customrs. Additionally, th dmand rat only dpnds on th lngth of a customr s crdit priod (i.. crdit-linkd dmand rat). Ouyang, Chang, and ng (005) gnralizd Goyal s (985) modl to obtain an optimal ordring policy for th rtailr whn th supplir provids not only a cash discount but also a prmissibl dlay in paymnts. ng, Ouyang, and Chn (006) dvlopd an conomic production quantity modl in which th manufacturr rcivs th supplir trad crdit and provids th customr trad crdit simultanously. In 007, Ouyang, Wu, and Yang proposd an conomic ordr quantity invntory modl with limitd storag capacity. hy considrd th situation whn h supplir provids a cash discount and a prmissibl dlay in paymnts for th rtailr. Ho, Ouyang, and Su (008) stablishd an intgratd supplir-buyr invntory modl with th assumption that th markt dmand is snsitiv to th rtail pric and discussd th trad crdit policy including a cash discount and dlayd paymnt. Chang, Ho, Ouyang, and Su (009) incorporatd th concpt of vndor-buyr intgration and ordr-siz-dpndnt trad crdit. hy prsntd a stylizd modl to dtrmin th optimal stratgy for an intgratd vndor-buyr invntory systm undr th condition of trad crdit linkd to th ordr quantity. Many rlatd paprs can b found in Hwang and Shinn (997), Jamal, Sarkr, and Wang (000), Liao, sai, and Su (000), ng (00), Abad and Jaggi (00), Arclus, Shah, and Srinivasan (00), Chang, Ouyang, and ng (00), Shinn and Hwang (00), Chang (00), Chung and Liao (00), ng (006), ng, Chang, Chrn, and Chan (007), ng and Goyal (007) and othrs. In th abov paprs, th rsarchrs adoptd th avrag cost approach, and did not considr th ffct of th tim-valu of it is ncssary to tak th ffct of th tim-valu of mony on th invntory policy into considration. rippi and Lwin (97) and Kim and Chung (990) rcognizd th nd to xplor invntory problms by using DCF approach or th prsnt valu concpt. Chung (989) adoptd th DCF approach for th analysis of th optimal invntory policy in th prsnc of a trad crdit. W and Law (00) dvlopd a dtriorating invntory modl with pric-dpndnt dmand and took into account th tim-valu of mony. hy applid th DCF approach for problm analysis. Grubbström and Kingsman (00) discussd th problm of dtrmining th optimal ordring quantitis of a purchasd itm whr thr ar stp changs in pric and th nt prsnt valu (NPV) principl is applid. W, Yu Jonas, and Law (005) stablishd a two warhous invntory modl with partial backordring and Wibull distribution dtrioration, in which th DCF and optimization framwork ar prsntd to driv th rplnishmnt policy that minimizs th total prsnt valu cost pr unit tim. Jaggi, Aggarwal, and Gol (006) prsntd th optimal invntory rplnishmnt policy for dtriorating itms undr inflationary conditions by using th DCF approach ovr a finit planning horizon. Dy, Ouyang, and Hsih (007) considrd an infinit horizon, singl product conomic ordr quantity whr dmand and h dtrioration rat ar continuous and diffrntiabl functions of pric and tim, rspctivly. hy applid th DCF approach to dal with th problm. Hsih, Dy, and Ouyang (008) stablishd an invntory modl for dtriorating itms with two lvls of storags, prmitting shortag and complt backlogging and usd th NPV of total cost as th obctiv function for th gnralizd invntory systm. Svral intrsting and rlvant paprs rlatd to trad crdits using th DCF approach ar Chapman, Ward, Coopr, and Pag (98), Dallnbach (986), Ward and Chapman (987), Jaggi and Aggarwal (99), Chung and Huang (000), Chung and Liao (006), Soni, Gor, and Shah (006), ng (006) and othrs. In this papr, w us th DCF approach to stablish an invntory modl for dtriorating itms with trad crdit basd on th ordr quantity. h supplir offrs his/hr customrs two altrnativs linkd to ordr quantity. On is that if th ordr quantity is gratr than or qual to a prdtrmind quantity, thn th supplir provids a longr prmissibl dlay in paymnts. Othrwis, th total purchas cost must b paid by a fixd crdit priod. h othr is that if th ordr quantity is gratr than or qual to a prdtrmind quantity, thn thr is a quantity discount on pric and th balanc must b paid by a fixd crdit priod. Othrwis, thr is no pric discount and th total purchas cost must b paid by a fixd crdit priod. W thn study th ncssary and sufficint conditions for finding th optimal solution to th problm. Finally, som numrical xampls ar prsntd to illustrat th thortical rsults. his articl is organizd as follows. In Sction, w provid th notation and assumptions. In Sction, w stablishd th discount cash-flow modls. Sction contains th 8

3 Optimal ordring policis using a discountd cash-flow thortical rsults. W obtain a thorm to show th optimal solution not only xists but also is uniqu for th prsnt valu functions of all futur cash outflows. h comparison among diffrnt policis is in Sction 5. Sction 6 provids a discussion of th diffrnt policy altrnativs basd on diffrnt paramtr valus. Svral numrical xampls ar givn to illustrat th rsults in this papr in Sction 7. Finally, Sction 8 draws conclusions and suggsts potntial dirctions for futur rsarch. Assumptions and Notations: h mathmatical modls in this rsarch articl ar dvlopd using following assumptions:. h dmand rat for th itm is stock dpndnt.. Shortags ar not allowd.. h rplnishmnt rat is instantanous.. im horizon is infinit. 5. h units in invntory dtriorat at a constant rat, 0. h dtrioratd units can nithr b rpaird nor rplacd during th cycl tim. 6. h supplir offrs his/hr customrs two altrnativs linkd to ordr quantity: (a) A prmissibl dlay. If th ordr quantity is gratr than or qual to a prdtrmind quantity Q, thn th total purchas cost must b paid by N. Othrwis, th total purchas cost must b paid by M (with N > M). (b) A pric discount. If th ordr quantity is gratr than or qual to a prdtrmind quantity Q, thn thr is a quantity discount on pric and th balanc must b paid by M. Othrwis, thr is no pric discount and th total purchas cost must b paid by M. In addition, th following notations ar usd throughout this papr: i : th out-of-pockt holding cost as a proportion of th itm valu pr unit tim C : th unit purchasing cost. A : th ordring cost pr ordr. r : th opportunity cost ( i.. th continuous discounting rat ) pr unit tim. Q : th ordr quantity Q : th minimum ordr quantity at which th prmissibl dlay in paymnts M is prmittd : th tim intrval such that Q units ar dpltd to zro du to both dmand and dtrioration Q : : th minimum ordr quantity at which th cash discount is availabl th tim intrval such that Q units ar dpltd to zro du to both dmand and dtrioration d : th cash discount rat, 0 < d < : th cycl tim (a dcision variabl). I t : th invntory lvl at any instant of tim t, 0 t M : th crdit priod at which th ordr quantity is lss than Q (or Q ) N : th crdit priod at which th ordr quantity is gratr than or qual to Q, with > N > M θ : constant rat of dtrioration, 0 θ R I t : th dmand rat at tim t. w considr RI t I t dnots rat of chang of dmand du to stock. PV() : th prsnt valu of cash flows for th first rplnishmnt cycl. () : th prsnt valu of all futur cash flows. Q* : th optimal ordr quantity. * : th optimal rplnishmnt tim intrval. * : th minimum prsnt valu of all futur cash flows, i.. * = (* ) Discountd cash flow mathmatical Modls: whr 0 is constant dmand and 0 9

4 Nita H. Shah and Amisha R. Patl h dpltion of th invntory is du to stock dpndnt dmand and dtrioration of units. h invntory lvl at any instant of is govrnd by th diffrntial quation: di t I t RI t, 0 t dt with th initial condition I(0) = Q and boundary condition I() = 0h solution of Eq. () is givn by, I t t, 0 t and ordr quantity is Q I 0 From (), w can obtain th tim intrval such that Q units ar dpltd to zro du to both stock dpndnt dmand and dtrioration as Q ln Consquntly, it is asy to s that th inquality Q < Q holds if and only if <. Similarly, Q ln and th inquality Q < Q holds if and only if <. Basd on th valus of, and, thr ar four cass to b considrd: () <, (), () () () () () (5) < and (). For ach cas, th corrsponding paymnt tim and cash discount rat ar prsntd in abl. abl. h paymnt tim and cash discount rat for diffrnt cass. Cas Paymnt tim Cash discount rat () N 0 Now, w dvlop th discountd cash-flow modl for ach cas. () < M 0 Cas. < (i.. th () M d ordr quantity is lss than Q ) h prsnt valu of cash flows for th first cycl; th componnts ar valuatd as following:. At th bginning of ach rplnishmnt cycl, cash out flows of th ordring cost, A. At th nd of th crdit priod, M, th customr pays th full purchas cost, CQ, if th ordr quantity is lss than Q. Using (), C Q C ; whr stands for th rplnishmnt tim intrval for cas.. h prsnt valu of th purchas cost at th continuous discounting rat r is C Q C rm () < M 0 0

5 Optimal ordring policis using a discountd cash-flow ic. h out of pockt invntory holding cost at tim t is t i C I t 5. h prsnt valu of th out of pockt holding cost in a rplnishmnt cycl with th continuous discounting ic rat r is rt t dt 0 Consquntly, th prsnt valu of cash flows for th first cycl is C rm ic rt PV A t dt 0 C rm ic PV A r r r r r hn th prsnt valu of all futur cash flows is PV PV nr PV nr n0 n0 r If M=0 and β = 0, thn Eq. (7) is th sam as Eq. (6) in Chung and Liao (006). If β = 0, thn Eq. (7) is as that of Chung t. al. (00) Cas. (i.. th ordr quantity is gratr than or qual to Q ) In this cas, th supplir offrs to customr a prmissibl dlay N in paymnts. By th similar procdur as dscribd in cas, w hav that th prsnt valu of all futur cash flows is PV PV nr PV nr n0 n0 r whr is th rplnishmnt cycl lngth for cas and C rn ic PV A r r r r r Cas. < (i.. th ordr quantity is lss than Q ) In this cas, th supplir offrs to customr a prmissibl dlay M in paymnts. hus th prsnt valu of all futur cash flows is PV PV nr PV nr n0 n0 r whr is th rplnishmnt cycl lngth for cas and C rm ic PV A r r r r r Cas. (i.. th ordr quantity is gratr than or qual to Q ) In this cas, th supplir offrs a prmissibl dlay M in paymnts also provids a cash discount with th rat d. hus th prsnt valu of all futur cash flows is PV PV nr PV nr n0 n0 r whr is th rplnishmnt cycl lngth for cas and C ( d ) rm ic ( d ) PV A r r r r r Combining th Eq. (7) - (0), th prsnt valu of all futur cash flows for ach cas is shown as follows: (6) (7) (8) (9) (0)

6 Nita H. Shah and Amisha R. Patl C( d ) ic( d ) A rm r r r r r r for =,,,, whr M = M = M = M and M = N, also d = d = d = 0 and d = d. Now, to dtrmin th optimal solution that provids th smallst prsnt valu of all futur cash flows for ach cas to purchasr.. hortical rsults In this sction, w dtrmin th optimal valu of (dnotd by ) for cas such that has a minimum valu, =,,,. By taking th first drivativ of in () with rspct to, w hav, d r ic( d ) r r r d r r whr and C ( d ) rm i C ( d ) A r C ( d ) rm i C ( d ) r Ltting d d 0, and rarranging trms, w obtain r i C ( d ) r r r From (), w can obtain th following rsult. horm 0, which satisfis Eq. () not only xists but also is uniqu, for =,,,. (a) h solution (b) h solution to () is th uniqu optimal valu that minimizs, for =,,, Proof. S Appndix for th dtaild proof. Now, w want to driv th xplicit closd-form solution of.utilizing th fact that r r r. as (β + θ + r) is small.. as (β + θ) is small. and (), w obtain r r i C ( d ) r A r r Consquntly, w hav th optimal rplnishmnt cycl tim * A A rm C d r i r Ltting M = M and d = 0, w obtain th optimal rplnishmnt cycl lngth for Cas as approximatly qual to * A C r rm i () () () () (5) (6)

7 Optimal ordring policis using a discountd cash-flow Obviously,, w substitut (6) into inquality, and obtain that if and only if A C r rm i h optimal prsnt valu of all futur cash flows can b obtaind as * rm i C A A C r i * r r r r Similarly, ltting M = N and d = 0, w obtain th optimal rplnishmnt cycl lngth for Cas as approximatly qual to * A C r rn i Substitut (9) into inquality, w obtain that (7) (8) (9) if and only if A C r rn i h optimal prsnt valu of all futur cash flows can b obtaind as * rn i C A A C r i * r r r r Likwis, lt M = M and d = 0, w obtain th optimal rplnishmnt cycl lngth for Cas as approximatly qual to * A C r rm i Substitut () into inquality, w obtain that if and only if A C r rm i h optimal prsnt valu of all futur cash flows can b obtaind as * rm i C A A C r i * r r r r Last, ltting M = M and d = d, w can obtain th optimal rplnishmnt cycl lngth for Cas as approximatly qual to * A C ( d) r rm i Substitut (5) into inquality, w obtain that if and only if A C ( d) r rm i (6) h optimal prsnt valu of all futur cash flows can b obtaind as * ( ) ( ) rm i C d A A C d r i * r r (7) r r 5. Comparisons among diffrnt policis: By comparing th optimal rplnishmnt cycls, ordrd quantitis and prsnt valus of all futur cash flows among ths four cass, w obtain th following rsult. (0) () () () () (5)

8 Nita H. Shah and Amisha R. Patl horm rm r i d r i rn (a) If and, thn Q Q Q Q rn r i d rm r i, thn (b) If Q Q Q Q rn r i d rm r i (c) If and and, thn Q Q Q Q Proof. rn r i d rm r i rm rn r i r i. hn, by Eq. (), Q If, thn from N > M, w obtain. Using Eq. (6), (9), () and (5), w can asily obtain, =,,, is an incrasing function of. hus, Q Q Q Q. his complts th proof of (a). By th similar procdur as dscribd in (a), th proofs of (b) and (c) can b asily compltd. 6. Comparisons among diffrnt policis hr ar thr situations: (A) Q Q ; (B) Q Q ; (C) Q Q for opting th optimal altrnativ basd on th valus of Q and Q. Now, w will talk about how to us th faturs of th optimal solution in ordr to find th optimal altrnativ asily. For notational convninc, Lt C r rm i, C r rn i, C r rm i and C ( d ) r rm i, Hnc w hav and. 6.. Situation A: Q Q his taks plac whn th minimum ordr quantity with a longr prmissibl dlay in paymnt N is largr than that with cash discount rat d. Furthrmor, w gt. Using Eq. (7), (0), () and (6) th rplnishmnt cycl * that minimizs th prsnt valu of all futur cash flows can b obtaind. hus, w hav th following rsults, for which th proof is trivial and hnc w omit it hr. horm. Whn Q Q Situation A Condition (*) * A min{ ( ), ( ), ( )} A min{ ( ), ( ), ( )} A min{ ( ), ( ), ( )}, or, or, or A min{ ( ), ( )} or A min{ ( ), ( ), ( )}, or A min{ ( ), ( ), ( ), ( )},, or

9 Optimal ordring policis using a discountd cash-flow A min{ ( ), ( )} or A min{ ( ), ( ), ( )},, or A ( ) 6.. Situation B: Q Q his occurs whn th minimum ordr quantity with a longr prmissibl dlay in paymnt Nis smallr than that with cash discount rat d, hnc,. By Eq. (7), (0), () and (6) th rplnishmnt cycl * that minimizs th prsnt valu of all futur cash flows can b obtaind and is givn as follows. h proof is omittd. horm. Whn Q Q Situation B Condition (*) * A min{ ( ), ( ), ( )}, or A min{ ( ), ( )} or A min{ ( ), ( )} or A min{ ( ), ( ), ( )}, or A ( ) A min{ ( ), ( )} or A min{ ( ), ( )} or A min{ ( ), ( ), ( )},, or A min{ ( ), ( ), ( ), ( )},, or A min{ ( ), ( )} or 6.. Situation C: Q Q his mans th minimum ordr quantity with a longr prmissibl dlay in paymnt N is th sam as that with cash discount rat d, thus, and thn. Using Eq. (7), (0),() and (6) th rplnishmnt cycl * that minimizs th prsnt valu of all futur cash flows can b obtaind and is givn as follows. h proof is omittd. horm 5. Whn Q Q Situation C Condition (*) * A min{ ( ), ( )} or A min{ ( ), ( ), ( )}, or A min{ ( ), ( )} or A min{ ( ), ( )} or 5

10 Nita H. Shah and Amisha R. Patl A ( ) 7. Numrical Exampls: Exampl. Considr, th paramtric valus α = 000 units, β = 0.05, C = 0, θ = 0.5, i = 0.,r = 0.05, d = 0.05, M = 5/65 yar = 5 days, N = 0/65 yar = 0 days. If Q 00 units and Q 80 units, w know that Q Q, , , 88.79, 88.08, , By using horm, w obtain th computational rsults for diffrnt valus of A as shown in abl. Exampl. By rplacing N = 5/65 yar = 5 days and Q 50 units, in Exampl, w obtain that Q Q, , 0.779, 88.79, 87.97, 96.6, By using horm, w obtain th computational rsults for diffrnt valus of A as shown in abl. Exampl. By substituting Q 00 units, in Exampl, w obtain that Q Q, , 88.79, 88.08, 88.79, By using horm 5, w obtain th computational rsults for diffrnt valus of A as shown in abl. Exampl. Rplacing A = 5 and Q 0 units, in Exampl, w obtain Q Q, 0.089, h computational rsults for diffrnt valus of d and N ar shown in abl 5 abl Optimal solution for diffrnt ordring costs A ( Q Q ) A * Q* * 0 y= = = = = abl Optimal solution for diffrnt ordring costs A ( Q Q ) A * Q* * 0 == == y= y= y= abl Optimal solution for diffrnt ordring costs A ( Q Q ) A * Q* * 0 y= y= = = = abl 5 Optimal solution for diffrnt cash discount rat d and N N d * Q* * =

11 Optimal ordring policis using a discountd cash-flow 0.05 = = = = = = = = = = = abl -6 Snsitivity analysis for α ( Q Q ) α A * Q* * y= = = = = y= = = = = y= = = = = abl -7 Snsitivity analysis for α ( Q Q ) α A * Q* * = = = y= y= == == y= y= y= = = y= y= y= abl -8 Snsitivity analysis for α ( Q Q ) α A * Q* * y= y= y=

12 Nita H. Shah and Amisha R. Patl 60 = = y= y= = = = y= y= = = = abl -9 Snsitivity analysis for β ( Q Q ) β A * Q* * y= = = = = y= = = = = y= = = = = abl -0 Snsitivity analysis for β ( Q Q ) β A * Q* * = = y= y= y= == == y= y= y= = = = y= y= abl - Snsitivity analysis for β ( Q Q ) β A * Q* * y= y=

13 Optimal ordring policis using a discountd cash-flow 5 = = = y= y= = = = y= y= = = = abl Snsitivity analysis for θ ( Q Q ) θ A * Q* * y= = = = = y= = = = = y= = = = = abl Snsitivity analysis for θ ( Q Q ) θ A * Q* * = y= y= y= y= ==

14 Nita H. Shah and Amisha R. Patl 5 == y= y= y= = = = y= y= abl Snsitivity analysis for θ ( Q Q ) θ A * Q* * y= = = = = y= y= = = = y= y= y= = = Conclusions: Appndix A o prov horm a, w st th lft-hand sid of () as r D r ; =,,,. aking th first drivativ of D with rspct to, w gt dd r r 0 d Hnc, D is a strictly incrasing function in, for =,,,. Furthrmor, D 0 r 0 ic( d) D r ( r) his complts th proof. that, and lim D. hrfor, thr xists a uniqu solution 0, such, that is, th solution 0, which satisfis Eq. () not only xists but also is uniqu. o prov horm b, w simply chck th scond-ordr condition at point. aking th scond drivativ of with rspct to and thn substituting into this quation, w obtain 0

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